Question

$prove \: that \sqrt{5} \: is \: irrational$
plzzz solve... ​

Answer 1

Hii mate..

Ur answer..

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let root 5 be rational

then it must in the form of p/q [q is not equal to 0][p and q are co-prime]

root 5=p/q

=> root 5 * q = p

squaring on both sides

=> 5*q*q = p*p  ------> 1

p*p is divisible by 5

p is divisible by 5

p = 5c  [c is a positive integer] [squaring on both sides ]

p*p = 25c*c  --------- > 2

sub p*p in 1

5*q*q = 25*c*c

q*q = 5*c*c

=> q is divisble by 5

thus q and p have a common factor 5

there is a contradiction

as our assumsion p &q are co prime but it has a common factor

so √5 is an irrational

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Answer 2

Answer:

let √5 is a rational number in form of p/q where p and q are the integer ans co-prime numbers.

√5= p/q

squaring on the both side

(√5)^2 = (p/q)^2

5=p^2/q^2

q^2=p^2/5

if 5 divide p^2 so it must be divide the p

let 5c=p

q^2=(5c)^2/5

q^2 =25c^2/5

q^2= 5c^2

q^2/5=c^2

if 5 divide the q^2 so it must be divide q also.

√5 is a irrational no. .our assumption is wrong beacause the p and q are

co-prime number and its not possible that 5 can divides the both...